How to style PopupMenu?

Yes, you can

<style name="YOURSTYLE.PopupMenu" parent="Widget.AppCompat.PopupMenu">
    <item name="android:textColor">@android:color/white</item>
    <item name="android:itemBackground">@android:color/holo_red_light</item>
</style>

And

Context wrapper = new ContextThemeWrapper(this, R.style.YOURSTYLE_PopupMenu);
PopupMenu popup = new PopupMenu(wrapper, view);

---

Result

You cannot set PopupMenu style directly, but there are other ways.

PopupMenu is created the following way:

PopupMenu popupMenu=new PopupMenu(context, anchorView);

The style of menu is determined by the style of context you pass. So all you need to do is to pass your Activity reference as context, and menu will be styled accordingly.

If you want to define the style yourself, inherit your activity style from one of the default ones and override the following items:

<style name="style" parent="android:Theme.Holo.Light">
    <item name="android:popupMenuStyle">...</item>
    <item name="android:popupAnimationStyle">...</item>
    <item name="android:popupBackground">...</item>
    <!-- etc etc -->
</style>

Adding to what Deville suggested, you can also add the following attributes to your theme style.

<style name="style" parent="android:Theme.Holo.Light">        
    <!-- other attributes -->
    <item name="textAppearanceLargePopupMenu">@style/myPopupMenuTextAppearanceLarge</item>
    <item name="android:textAppearanceLargePopupMenu">@style/myPopupMenuTextAppearanceLarge</item>

    <item name="textAppearanceSmallPopupMenu">@style/myPopupMenuTextAppearanceSmall</item>
    <item name="android:textAppearanceSmallPopupMenu">@style/myPopupMenuTextAppearanceSmall</item>

    <item name="popupMenuStyle">@style/myPopupMenuStyle</item>
    <item name="android:popupMenuStyle">@style/myPopupMenuStyle</item>
</style>

Other styles referenced in the above style definition

<style name="myPopupMenuStyle" parent="@style/Widget.AppCompat.Light.PopupMenu">

</style>
<style name="myPopupMenuTextAppearanceSmall" parent="@style/TextAppearance.AppCompat.Light.Widget.PopupMenu.Small">
    <item name="android:textColor">#000000</item>
</style>
<style name="myPopupMenuTextAppearanceLarge" parent="@style/TextAppearance.AppCompat.Light.Widget.PopupMenu.Large">
    <item name="android:textColor">#000000</item>
</style>

You would notice AppCompat in my xml style definitions, that is because I am using the android support library to target lower android API Levels.